update?

Jenkinsfile

@@ -1,6 +1,6 @@
 pipeline {
     environment {
-        registry = "trance0/NoteNextra"
+        registry = "trance0/notenextra"
         version = "1.0"
     }
     

docker-compose.yaml

@@ -3,7 +3,7 @@ services:
     build:
       context: ./
       dockerfile: ./Dockerfile
-    image: trance0/notenextra:v1.1.5
+    image: trance0/notenextra:v1.1.6
     restart: on-failure:5
     ports:
       - 13000:3000

pages/CSE332S/_meta.js

@@ -16,4 +16,5 @@ export default {
     CSE332S_L11: "Object-Oriented Programming Lab (Lecture 11)",
     CSE332S_L12: "Object-Oriented Programming Lab (Lecture 12)",
     CSE332S_L13: "Object-Oriented Programming Lab (Lecture 13)",
+    CSE332S_L14: "Object-Oriented Programming Lab (Lecture 14)",
 }

pages/Math4121/Math4121_L21.md

@@ -1 +1,74 @@
-# Lecture 21
+# Math4121 Lecture 21
+
+## Rolling from last lecture
+
+### Convergence of integrals
+
+#### Arzela-Osgood Theorem
+
+Let $\{f_n\}$ be a sequence of function, $f(x)=\lim_{n\to\infty}f_n(x)$ for every $x\in [0,1]$, if $f\in \mathscr{R}[0,1]$, and $\exists B>0$ such that $|f_n(x)|\leq B \forall x\in [0,1]$. (uniformly bounded and integrable)
+
+$$
+\lim_{n\to\infty}\int_0^1 f_n(x) dx = \int_0^1 f(x) dx
+$$
+
+If we let $\Gamma_{\alpha}$ be the set of intervals where $f_n$ is not continuous,
+
+$$
+\Gamma_{\alpha} = \{x\in [0,1] : \textup{ for any }m\in \mathbb{N}, \delta > 0, \exists n\geq m, y\in (x-\delta, x+\delta) \text{ s.t. } |f_n(y)-f(y)|>\alpha\}
+$$
+
+Fact: $\Gamma_{\alpha}$ is closed and nowhere dense.
+
+Proof:
+
+Without loss of generality, we can assume $f=0$. Given any $\alpha > 0$, $\exists N$ such that
+
+$$
+\left|\int_0^1 f_n(x) dx \right| < \alpha
+$$
+
+for all $n\geq N$.
+
+Consider the set $\Gamma_{\alpha/2} = \bigcup_{n=1}^{\infty} E_n$, for each $g\in \Gamma_{\alpha/2}$, we still have $\lim_{n\to\infty}f_n(g) = 0$.
+
+So we define
+
+$$
+G_i=\{g\in \Gamma_{\alpha/2} :|f_n(g)|<\frac{\alpha}{2} \text{ for all }n\geq i\}
+$$
+
+So $G_1\subset G_2\subset \cdots$ and $\Gamma_{\alpha/2} = \bigcup_{i=1}^{\infty} G_i$.
+
+By Osgood Lemma, since $\Gamma_{\alpha/2}$ is closed, $\exists K$ such that $c_e(G_K)>c_e(\Gamma_{\alpha/2})-\frac{\alpha}{4B}$.
+
+By definition of $c_e$, we cna find open $I_1,\ldots,I_N$ which cover $\Gamma_{\alpha/2}$ and
+
+$$
+\sum_{i=1}^N \ell(I_i) < c_e(\Gamma_{\alpha/2})+\frac{\alpha}{4B}
+$$
+
+Let $\mathcal{U}=\bigcup_{i=1}^N I_i$, and $\mathcal{C}=[0,1]\setminus \mathcal{U}$.
+
+Part 1: Control the integral on $\mathcal{C}$
+
+for each $x\in \mathcal{C}$, $x\notin \Gamma_{\alpha/2}$, so $\exists$ and open interval $I(x)$ and an integer $m(x)$ such that $|f_{m(x)}(x)|<\frac{\alpha}{2}$ and $\forall n\geq m(x), y\in I(x)$
+
+So $\mathcal{C}\subset \bigcup_{x\in \mathcal{C}} I(x)$, and $\mathcal{C}$ is closed and bounded, $\exists x_1,\ldots,x_J$ such that $\mathcal{C}\subset \bigcup_{j=1}^J I(x_j)$. So if $n\geq \max_{j=1,\ldots,J} m(x_j)$, and $x\in \mathcal{C}$, then $|f_n(x)|<\frac{\alpha}{2}$.
+
+So $\int_\mathcal{C} |f_n(x)| dx < \frac{\alpha}{2} c_e(\mathcal{C})$.
+
+Part 2: Control the integral on $\mathcal{U}$
+
+If $[x_i,x_{i+1}]\cap G_k\neq \emptyset$, then $\inf_{x\in [x_i,x_{i+1}]} |f_n(x)| < \frac{\alpha}{2}$ for all $n\geq K$. Denote such set as $P_1$.
+
+Otherwise, we denote such set as $P_2$.
+
+So $\ell(\mathcal{U})=\ell(P_1)+\ell(P_2)\geq c_e(G_K)+\ell(P_2)$.
+
+This implies $\ell(P_2)\leq \frac{\alpha}{4B}$.
+
+Continue on Friday.
+
+QED
+

pages/Math4121/_meta.js

@@ -22,15 +22,9 @@ export default {
     Math4121_L17: "Introduction to Lebesgue Integration (Lecture 17)",
     Math4121_L18: "Introduction to Lebesgue Integration (Lecture 18)",
     Math4121_L19: "Introduction to Lebesgue Integration (Lecture 19)",
-    Math4121_L20: {
+    Math4121_L20: "Introduction to Lebesgue Integration (Lecture 20)",
-        display: 'hidden'
+    Math4121_L21: "Introduction to Lebesgue Integration (Lecture 21)",
-    },
+    Math4121_L22: "Introduction to Lebesgue Integration (Lecture 22)",
-    Math4121_L21: {
-        display: 'hidden'
-    },
-    Math4121_L22: {
-        display: 'hidden'
-    },
     Math4121_L23: {
         display: 'hidden'
     },